Second derivative tests

Second derivative tests: Let $f:I\to\mathbb{R}$ be twice differentiable on an open interval $I$, and let $c\in I$ satisfy $f'(c)=0$.

• If $f''(c)>0$, then $c$ is a strict local minimum of $f$.
• If $f''(c)<0$, then $c$ is a strict local maximum of $f$.
• If $f''(c)=0$, no conclusion follows from this test alone.

These criteria refine the notion of a critical point and provide a standard way to detect a local extremum using higher derivatives.